First Chern class of toric manifolds

Question

I have been reading a Mirror Symmetry monograph, and its physical arguments seem to imply that all toric manifolds have semipositive-definite first Chern class.

Is this true, and if yes, how does one show this rigorously?

Solution from physics:

In Chapter 7 (page 102) of the Mirror Symmetry monograph, it is said that "Toric varieties can be described as the set of ground states of an appropriately gauged linear sigma model (GLSM)". However, from Chapters 14 and 15 of the same book, it can be deduced that the GLSM can only provide a description of toric manifolds $X$ with $c_1(X)\geq0$ (the reason is roughly that nonlinear sigma models for $X$ with $c_1(X)<0$ are not well-defined).

Thank you.

Answer 1

No, this is not true. The simplest examples come from Hirzebruch surfaces, as discussed in Chapter 7 of the linked monograph. These are smooth projective toric surfaces obtained by projectivising rank-2 bundles over $\mathbf P_1$, so they look like $F_n = \mathbf P(O \oplus O(n))$ for some natural number $n$. One can check that for $n \geq 3$ this does not have semipositive first Chern class.